%I M2537 #103 Sep 08 2022 08:44:33

%S 4,3,6,10,21,46,108,263,658,1674,4305,11146,28980,75547,197262,515594,

%T 1348477,3528150,9233244,24167167,63261114,165604618,433534041,

%U 1134967250,2971318756,7778909811,20365282518,53316730378,139584573093,365436446014,956723886540

%N a(n) = 1 + L(n) + F(2*n-1) with {L(n)}_{n>=0} the Lucas numbers (A000032) and F(2*n-1)_{n>=0} the bisected Fibonacci numbers (A001519).

%C From _Russell Jay Hendel_, Mar 02 2015: (Start)

%C The Name field was changed. The former Name-field entry was:

%C a(n) = Fibonacci(2n)*(1 + Fibonacci(n-1))/Fibonacci(n) for n even; a(n) = 2 + Fibonacci(2n)*(1 + Fibonacci(n-1))/Fibonacci(n) for n odd.

%C The new entry in the Name field is based on Equation (1) of the Grieg paper cited in the Links section. This formula is simpler than the former formulas given in the Name and Formula fields of this sequence. We make 6 comments:

%C (Comment I) The former formula in the Name section is consistent with a(0)=4 since 0 is a removable singularity of the former formula in the Name section. To see this we use the continuous Binet form, F(x) = (alpha^x - beta^x)/sqrt(5) with alpha = (sqrt(5)+1)/2 and beta = 1-alpha. Hence, for all nonzero x, F(2x)/F(x) = (alpha^(2x) - beta^(2x))/(alpha^x - beta^x) = alpha^x + beta^x. It immediately follows that lim_{x->0} F(2x)/F(x) = alpha^0 + beta^0 = 2. Consequently, lim_{x->0} Fibonacci(2x)*(1 + Fibonacci(x-1))/Fibonacci(x) = 2(1+Fibonacci(-1)) = 4 as required.

%C (Comment II) As pointed out in references [2,3] of the Grieg paper, a(n) = F(2n) C(n), with C(n) = Sum_{m>=0} 1/F(n*b), b=2^m, and {F(n)}_{n>=0}, the Fibonacci numbers (A000045). In other words, the a(n) = C(n) F(2n) are the numerators of the sequence of C(n) = a(n) / F(2n), which are the values of infinite sums of reciprocals of subsets of the Fibonacci numbers.

%C (Comment III) The probable reason that the elegant formulas in the Grieg papers were overlooked is that the Grieg papers are difficult to read because of their nonstandard notation and terminology. For example, the generalized Fibonacci and Lucas numbers are indicated by P with subscripts rather than by F. Similarly, Grieg refers to the sequence of odd-indexed Fibonacci numbers as the "bisected" Fibonacci numbers (A001519), a term which rarely appears in modern books on the Fibonacci numbers.

%C (Comment IV) The defining recursion for a(n) is best given in factored form using annihilating operators (Equation (12) in the Grieg paper). Let E be the translation operator so that E(f)(n) = f(n+1). Then, for example, (E^2-E-1)(L(n)) = L(n+2) - L(n+1) - L(n) = 0 (A000032). Thus the operator E^2-E-1 annihilates the Lucas sequence. Similarly, (E^2-3E+1) annihilates the bisected Fibonacci numbers (A001519) and (E-1) annihilates the constant sequence (A000012). It follows that the product of these annihilators annihilates the sum of the sequences, which is a(n). So the defining recursion may be obtained by expanding (E-1)(E^2-3E+1)(E^2-E-1). It follows that the sequence a(n) satisfies a(n) - 5 a(n-1) + 7 a(n-2) - a(n-3) - 3 a(n-4) + a(n-5) = 0. For example, 1*108 - 5*46 + 7*21 - 1*10 - 3*6 + 1*3 = 0. Note that the derivation of this recursion, based on annihilators, is more straightforward than the derivation in the Grieg paper based on central difference operators.

%C (Comment V) Using standard techniques, we may obtain the g.f. of the a(n) from the defining recursion, of order 5. Since a(n) = 1 + F(2n-1) + L(n), it follows that the g.f. for the a(n) is the product of the g.f. for the three summands 1, F(2n-1), L(n). The g.f. for the constant sequence, 1, is C(x)=1/(1-x) (A000012); the g.f. for the Lucas sequence is L(x)=(2-x)/(1-x-x^2)(A000032); and the g.f. for the bisected Fibonacci numbers F(2n-1) is B(x)=(1 - 2x)/(1 - 3x + x^2) (sequence A001519). It follows that the g.f. for the a(n) is the sum of these g.f.s: C(x) + L(x) + B(x) = 1/(1-x) + (1-2x)/(1-3x+x^2) + (2-x)/(1-x-x^2). This is a partial fraction decomposition of a rational function, that is an equivalent g.f. for the a(n), A(x) = (4 - 17x + 19x^2 - 3x^3 - 2x^4)/(1 - 5x + 7x^2 - x^3 - 3x^4 + x^5).

%C (Comment VI) We can now comment on the g.f., P(x), conjectured by _Simon Plouffe_ and mentioned in the MAPLE section below. Expanding P(x) as a series, we obtain the following g.f. of the a(n): 3 + 6x + 10x^2 + 21x^3 ... . In fact, it is easily checked that P(x) = 1/x (A(x)-4). This settles the Plouffe conjecture as follows: (i) A(x) is the rational function g.f. of a(n)_{n>=0}, (ii) P(x) is the rational function g.f. of a(n)_{n>=1}, and (iii) C(x)+L(x)+B(x) is the partial fraction g.f. of a(n)_{n>=0}. (End)

%C Let phi = (1+sqrt(5))/2, p(n) = phi^n - (-phi)^(-n) and FL(n) = 1 + (p(n-1) + p(n+1) + p(2*n-1))/sqrt(5). FL is a doubly infinite sequence with FL(n) = A005522(n) and FL(-n) = A006172(n) for n >= 0. - _Peter Luschny_, Mar 09 2015

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A005522/b005522.txt">Table of n, a(n) for n = 0..1000</a>

%H W. E. Greig, <a href="https://www.fq.math.ca/Scanned/15-1/greig2.pdf">Sums of Fibonacci Reciprocals</a>, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 46-48.

%H W. E. Greig, <a href="http://www.fq.math.ca/Scanned/16-2/greig.pdf">On generalized G_{j,k} numbers</a>, Fib. Quart., 16 (1978), 166-170.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,1,3,-1).

%F a(n) = ((sqrt(5)+5)/10)*(((3+sqrt(5))/2)^(n-1)) + ((5-sqrt(5))/10)*(((3-sqrt(5))/2)^(n-1)) + ((1+sqrt(5))/2)^n + ((1-sqrt(5))/2)^n + 1. - _Tim Monahan_, Jul 23 2011

%F a(n) = 1 + L(n) + F(2*n-1), with L(n) the Lucas numbers, sequence A000032 and F(2n-1) the bisected Fibonacci numbers, sequence A001519. - _Russell Jay Hendel_, Mar 02 2015

%F a(n) = 5*a(n-1) - 7*a(n-2) + a(n-3) + 3*a(n-4) - a(n-5). - _Russell Jay Hendel_, Mar 02 2015

%F Sum_{n>=1} (a(2*n-1)/Fibonacci(4*n-2) - 1/phi) = A079586, where phi is the golden ratio (A001622) (Greig, 1977). - _Amiram Eldar_, Jan 29 2022

%p with(combinat): a:=proc(n) if n mod 2 = 0 then fibonacci(2*n)*(1+fibonacci(n-1))/fibonacci(n) else 2+fibonacci(2*n)*(1+fibonacci(n-1))/fibonacci(n) fi end: seq(a(n),n=1..32); # _Emeric Deutsch_, Apr 01 2005

%p A005522:=-(-3+9*z-z**2-10*z**3+4*z**4)/(z-1)/(z**2-3*z+1)/(z**2+z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%p FL := proc(n) local a,p; a := (1+sqrt(5))/2; p := m -> a^m - (-a)^(-m);

%p 1 + (p(n-1) + p(n+1) + p(2*n-1))/sqrt(5) end: A005522 := n -> FL(n):

%p seq(round(evalf(A005522(n),32)), n=0..30); # _Peter Luschny_, Mar 09 2015

%t Table[1+Fibonacci[2n-1]+LucasL[n]), {n, 0, 30}] (* _Russell Jay Hendel_, Mar 02 2015 *)(* modified by _G. C. Greubel_, Jan 01 2018 *)

%t Series[1/(1-x)+ (2-x)/(1-x-x^2)+(1-2*x)/(1-3*x+x^2), {x, 0, 30}] (* _Russell Jay Hendel_, Mar 02 2015 *)

%t LinearRecurrence[{5,-7,1,3,-1}, {4, 3, 6, 10, 21}, 31] (* _Russell Jay Hendel_, Mar 02 2015 *)

%t CoefficientList[ Series[(4 - 17x + 19x^2 - 3x^3 - 2x^4)/(1 - 5x + 7x^2 - x^3 - 3x^4 + x^5), {x, 0, 30}], x] (* _Russell Jay Hendel_, Mar 02 2015 *)

%o (PARI)

%o \ps {31};

%o C(x) = 1/(1-x);

%o L(x) = (2-x)/(1-x-x^2);

%o B(x) = (1-2*x)/(1-3*x+x^2);

%o A(x) = C(x)+L(x)+B(x);

%o Ser(A(x),x)

%o Vec(Ser(A(x),x))

%o \\ _Russell Jay Hendel_, Mar 02 2015

%o (Magma) [1 + Lucas(n) + Fibonacci(2*n - 1): n in [0..30]]; // _Vincenzo Librandi_, Mar 08 2015

%o (PARI) {lucas(n) = fibonacci(n+1) + fibonacci(n-1)};

%o for(n=0,30, print1(1 + fibonacci(2*n-1) + lucas(n), ", ")) \\ _G. C. Greubel_, Jan 01 2018

%Y Cf. A000032, A000045, A001519, A001622, A006172, A079586, A254884.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

%E Edited by _Emeric Deutsch_, Apr 01 2005

%E a(0), a(29), a(30) added by _Russell Jay Hendel_, Mar 02 2015

%E Name changed by _Russell Jay Hendel_, Mar 02 2015